Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12310 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#12310 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
1090 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 1090 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(4,8)(5,12)(6,14)(9,13)(10,17)(11,16)(15,18);(5,12)(6,14)(8,13)(9,11)(15,18)(16,17);(3,4)(5,13)(6,9)(7,10)(8,12)(11,14)(15,16)(17,18)68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11711452, #72213969, #72214134, … |
| maximal normal subgroup of | #72213969, #72214134, #72214250, … |
| maximal quotient of | C2×#143196, C2×#125647, C2×#125721, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396061);